Bundles, curvature and holonomy

Thirty-four chapters develop bundles, connections, curvature, holonomy, Riemannian and Kähler geometry, and integral characteristic classes. Complete proofs and exact earlier chapter links accompany the results.

These thirty-four chapters are self-checked by the writing AI.

Checked chapters

1. Local tools for bundles and transport
2. Integral Thom classes and Euler indices
3. Manifold duality and the Euler characteristic
4. Projective bundles, Gysin sequences and line classes
5. Chern classes and the Whitney formula
6. Curvature, characteristic classes and Gauss–Bonnet
7. Principal bundles and associated bundles
8. Connections and parallel transport
9. Curvature and holonomy groups
10. Reduction and the holonomy theorem
11. Flat connections and infinitesimal holonomy
12. Invariant connections on homogeneous bundles
13. Linear and affine connections
14. Geodesics, normal coordinates and curvature
15. Riemannian connections and convex neighbourhoods
16. Completeness and the Hopf–Rinow theorem
17. Holonomy and the de Rham decomposition theorem
18. Sectional curvature and space forms
19. Affine transformations and the isometry group
20. Holonomy, Killing fields and analytic extension
21. Submanifolds and hypersurfaces
22. Jacobi fields, conjugate points and the Morse index theorem
23. Comparison theorems, cut loci and curvature and topology
24. Complex manifolds and Kähler metrics
25. Kähler curvature and Hermitian vector bundles
26. Homogeneous spaces and invariant connections
27. Symmetric spaces
28. Symmetric Lie algebras and Hermitian symmetric spaces
29. Characteristic classes and the Weil homomorphism
30. Vector bundles and their constructions
31. Fundamental classes from finite chart gluing
32. Grassmannians and classifying maps
33. Schubert cells and universal Grassmannian cohomology
34. Pontryagin classes and the Euler square

Human construction sources and component terms remain in the chapters. Rendering software and fonts retain their notices.